Cascade of failures in coupled network systems with multiple support-dependence relations
We study, both analytically and numerically, the cascade of failures in two coupled network systems A and B, where multiple support-dependence relations are randomly built between nodes of networks A and B. In our model we assume that each node in one network can function only if it has at least a single support link connecting it to a functional node in the other network. We assume that networks A and B have (i) sizes ${N}^{A}$ and ${N}^{B}$, (ii) degree distributions of connectivity links ${P}^{A}(k)$ and ${P}^{B}(k)$, (iii) degree distributions of support links ${\mathrm{P\ifmmode \tilde{}\else \~{}\fi{}}}^{A}(k)$ and ${\mathrm{P\ifmmode \tilde{}\else \~{}\fi{}}}^{B}(k)$, and (iv) random attack removes $(1\ensuremath{-}{R}^{A}){N}^{A}$ and $(1\ensuremath{-}{R}^{B}){N}^{B}$ nodes form the networks A and B, respectively. We find the fractions of nodes ${\ensuremath{\mu}}_{\ensuremath{\infty}}^{A}$ and ${\ensuremath{\mu}}_{\ensuremath{\infty}}^{B}$ which remain functional (giant component) at the end of the cascade process in networks A and B in terms of the generating functions of the degree distributions of their connectivity and support links. In a special case of Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi networks with average degrees $a$ and $b$ in networks A and B, respectively, and Poisson distributions of support links with average degrees $\mathrm{a\ifmmode \tilde{}\else \~{}\fi{}}$ and $\mathrm{b\ifmmode \tilde{}\else \~{}\fi{}}$ in networks A and B, respectively, ${\ensuremath{\mu}}_{\ensuremath{\infty}}^{A}={R}^{A}[1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\mathrm{a\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\mu}}_{\ensuremath{\infty}}^{B})][1\ensuremath{-}\mathrm{exp}(\ensuremath{-}a{\ensuremath{\mu}}_{\ensuremath{\infty}}^{A})]$ and ${\ensuremath{\mu}}_{\ensuremath{\infty}}^{B}={R}^{B}[1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\mathrm{b\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\mu}}_{\ensuremath{\infty}}^{A})][1\ensuremath{-}\mathrm{exp}(\ensuremath{-}b{\ensuremath{\mu}}_{\ensuremath{\infty}}^{B})]$. In the limit of $\mathrm{a\ifmmode \tilde{}\else \~{}\fi{}}\ensuremath{\rightarrow}\ensuremath{\infty}$ and $\mathrm{b\ifmmode \tilde{}\else \~{}\fi{}}\ensuremath{\rightarrow}\ensuremath{\infty}$, both networks become independent, and our model becomes equivalent to a random attack on a single Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi network. We also test our theory on two coupled scale-free networks, and find good agreement with the simulations.